cheesemonkey wonders

cheesemonkey wonders

Wednesday, April 27, 2016

"De-tracking" Versus Mastery: Is This Our Dirtiest Little Secret...?

There has been so much heat and noise (and not very much light) on all sides of the so-called "de-tracking" debate, it has made me want to raise a question I have been thinking a lot about:
What is the difference between "tracking" (i.e., ability grouping, as in "high-," medium," or "low") and an SBG-style measure of mastery?
I ask because as someone who thinks about classroom instruction in a deeply Vygotskian way, I value the Zone of Proximal Development (ZPD) above almost all else in figuring out how to ensure that all my students receive meaningfully differentiated instruction.

But if there IS no reasonably common ZPD, there's no way I can see to differentiate — apart from simply allowing everybody to work at their own pace... in which case, what good am I in the room?

How do you make sense of this distinction?

Wednesday, April 6, 2016

Volume of a Pyramid: Proof by Play-Doh

This is the best idea I never had.

My colleague, Tom Chan, asked me in the Math Office this morning, "Where are you guys at?"

I told him, "We're on volume of a pyramid."

"Me too!" He's usually a pretty cool cucumber, so this caught me by surprise. He said, "We're doing proof of the volume formula by Play-Doh. Wait here a minute."

He dashed out and came back within a minute with a fist-sized cube made of three different colors of Play-Doh.

"Each table gets three little tubs (so three colors) and they have to make three identical pyramids that fit together into a cube. Then they can move on and do the next piece."

I was dumbfounded. The best I'd been able to do for today was to produce tiny, helpful diagram handouts to fit into our INBs.

But I'm bookmarking this for myself for next year by blogging it, and by giving full credit.

Thursday, March 24, 2016

Algebra 1 quadratics — which method and why

When kids demonstrate that they don't yet have a solid-enough fluency to move on to deliberate practice with metacognitive reflection, it's time to go back to the drawing board.

That's what happened this morning with my second-block Algebra 1 class.

So during third block, I went to the library and started fishing in the MTBoS Search Engine. I wanted a card sort activity or an idea for one.

It didn't take long to find out that Dane Ehlert and Geoff Krall had already come to the same conclusion independently — and that they had even done some of the work for me!

Everybody's kids are at different levels when you slam into a new topic. So it's great to be able to find the structure of an activity that you can easily adapt to fit your own students' actual depth of knowledge.

This is why, even though I love a lot of the Shell Centre activities, I often find that MTBoS adaptations (or my own) are best for the reality of my classroom. They've given us some fantastic models to use in our actual teaching and learning.



Monday, March 21, 2016

What to do when strong students struggle

Jessica Lahey just posted this column on the New York Times web site and I think it may be the most important read I have seen for parents of the kinds of students I teach:

All students encounter struggle. Even strong students struggle. And when this happens, parents often ask me how they can make their child care more about doing better in math.

The only answer I know — the only answer I trust — is that you have to be willing to allow them to struggle.

Only then can they truly own their own success.

If they don't own their own failure, then they can't own their own success.

The eminent child psychologist Rudolf Dreikurs wrote about this more than 50 years ago, and it is as true now as it was then. You have to step back and let them own it. Dreikurs called this the practice of using natural and logical consequences. If the child doesn't own the problem, then s/he cannot own the solution.

I also love Dr. Charlotte Kasl's framing of this. She calls it the "Good luck with that!" response. I have seen this approach be very successful with students who have internalized a kind of passivity or learned helplessness that drives adults crazy. They have learned how to get adults to rescue them.

I think of this not as "tough love" or "grit" or a growth mindset. I think this is about the practice of maintaining — and helping adolescents learn how to maintain — strong, healthy boundaries.

Friday, March 18, 2016

Let the kids teach themselves how to do Talking Points

In thinking about the ways in which I try to push authority downward into student groups, I have been searching for ways to get my students to teach themselves about how to do Talking Points.

scene from the forthcoming Harry Potter and the Chapter on Inequalities
from the forthcoming mathematical
blockbuster, Harry Potter and the 
Chapter on Inequalities
So far, this has been by far my most successful method.

I have written "deleted scenes" from unmade (or yet-to-be-made) movies to introduce new concepts by having kids do a readers' theater activity instead of lecturing. So, I thought, why not do the same thing for Talking Points?

The results have been much better than I expected. Because all the voices and rules come to them through their own voices, they seem much more bought into the guidelines. They also act as their own enforcers of norms, rather than my having to circulate around the room constantly on the lookout for infractions.

So here is a link to my deleted scene for having kids teach themselves about Talking Points.

I will also be using this scene in my NCTM workshop on Talking Points next month.

Let me know what you think!

Saturday, February 6, 2016

Lessons from "Lessons from Bowen and Darryl"

I had the beginnings of a blinding insight this week that I wanted to write down and think about. It all started with Ben Blum-Smith’s blog post about what he learned from Bowen and Darryl’s master class at JMM earlier this year. He wrote up his takeaways and it sparked many great Twitter conversations.

I wanted to write about what *I* took away from his post and what I have learned from subsequent Twitter conversations with Bowen and Darryl.

My burning question this week that I posed for them after reading Ben’s post was, how do they manage the mixture of “speed demons” and “katamari” in their class work. This distinction between speed demons and katamari was really obvious once I read about it; yet, it remains the dirty little secret of math pedagogy — and the unspoken, problematic truth of group work strategies.

Both of these approaches—the “speed demon” approach and the “katamari” approach —  are highly personal, highly developed inner world views within mindsets. They are not labels or designations applied from without. They are basic “come-from” attitudes that arise from within.

I have my own truth of this distinction from my own experiences, and I believe that we all approach it with our own biases and theoretical frameworks. I freely admit that I speak as a katamari with a lifetime of bad experiences in mathematical group work, both with fellow students and with professional colleagues. In mathematics and in group work, I find that I still experience these hidden assumptions over and over again: what I perceive as the tyranny of the speed demons and my own resigned sense of hopelessness that my tortoise-like katamari learning style — including my tortoise-shell defense mechanisms against the feelings of rage and powerlessness and inner worthlessness as a math learner I experience whenever I am asked to do mathematical learning in a group with others.
[EDITORIAL NOTE: Please don’t worry that you need to rescue me. Or that I need you to rescue me. I don’t and you don’t. These are only thoughts flowing down the river of mind, and I have learned how to notice them and work with them through many years of self-noticing, meditation, inner development work, and therapy. I’m not held back by them, and I don’t need to be reassured about them. But the reality of inner development is that those deep-rooted holes and feelings don’t go away. We just learn how to notice them and work with them more skillfully and artfully so we can continue under all circumstances. Over time, they lose their power. They just become the voice of monkey mind yakking away in the background. And I have learned how to work with that noise in the background and to tune it out].
For math learners with a high degree of natural aptitude and curiosity, the speed demon world view is a very natural attitude to develop. You love math, you want more of it, and you are totally and completely voracious. I see this in many of my students. They want to devour math. They’re not just hungry, they are  completely driven by their appetite for math. More more more. Faster faster faster. Nom nom nom. They find math delicious, and they want to eat all they can as fast as they can. They want to climb every mountain they encounter. The higher, the better. More mountains, please. More math.

They’re joyful — they’re not intentionally being aggressive in the classroom. But they are children. They don’t have a lot of self-control or self-regulation skills, and they’re adolescents, so they don’t have a lot of awareness of how others are feeling in the moment. That is why they have to be managed and fed in the classroom learning process.

Meanwhile, the katamari in my classes are experiencing things quite differently. But first, we should mention that there is a HUGE range of ability, interest, and aptitude among the katamari. In fact, the population of katamari is where the greatest range of learners exists. But they are distinguished by their katamari worldview, which arises in binary opposition to the speed demon worldview. They work through things at their own pace and they alternate between individual think time moments and collaboration. And they build as patiently as they can on what they figure out.

Bowen and Darryl propose a revolutionary approach to managing the mixture of speed demons and katamari within the problem-based, group-work-centric math class. They divide classroom time into “doing” segments and whole-class “discussing” segments.

Here is my summary understanding of the three key parts of their deployment strategy for the problem-based learning experience.
1. Problem sets are designed to be a treasure map — but the essential treasure in any day’s work is located within the “Important Stuff” initial section
2. Groupings must facilitate individual discovery together — and they must eliminate all social and emotional obstacles to including *everybody* in the process of discovery See #1.
3. Whole-class discussion segments exist only to feature the discoveries that katamari have made by slowing down and really noticing the often subtle and deep mathematics that can be noticed through careful work
Some elaboration on all of this:


The daily problem set is designed as a treasure map, but the secret is that all essential mathematics in any day’s work is located in the first section, not at the end of the problem set.

Interesting Stuff and Tough Stuff exist to provide nourishment for anyone who is ready to explore it. But it does NOT contain the keys to the kingdom. If speed demons wish to zoom ahead and tackle the “tough stuff” that is there to satisfy their appetite for zooming into zoomy, zoomy heights, then they are welcome to do so. Meanwhile, the katamari can find reassurance in the fact that their worldview and their approach is explicitly being valued.


This point is key. Our students are still adolescents, and they are starting from a place of little impulse control when it comes to their own self-interest. Since the only way to develop intrinsic motivation is through autonomy, mastery, and purpose, we need to tap into that rather than try to manipulate everybody into clamping down on their natural orientations.

Since what motivates people is a combination of autonomy, mastery, and purpose (see Dan Pink, Drive), our grouping strategies in the classroom HAVE TO support student autonomy. In other words, if speed demons believe they’ve gotta speed, then our groupings need to support their desire/need for speed.

It’s important to separate the speed demons from the katamari, but there is a crucial misunderstanding as to why. Many people believe that this places an undue burden on the speed demons, but that’s actually backwards from the truth. The reality is, mixing the groups during the “doing” segments actually places an unfair burden on the katamari. It requires them to allow the speed demons to dominate the learning process, to be the center of attention at all times, and to cheat them out of their understanding.

So for this reason, it’s important to let the speed demons go off in zoomy groups and zoom away. This is not the place where they are going to learn the social and emotional interpersonal skills they need because it’s NOT the place where they are receptive to these lessons. Instead, this is the place where we, the adults, need to create safe space for katamari to work at their own pace and to develop the learnings they need in order to move forward.

*This* is meaningful differentiation.

So during the doing segments, I am now going to let the speed demons zoom. In fact, I am going to set up my speed demons so they can go off and do their zoomy zoomy zoom investigations with the “Interesting Stuff” and the “Tough Stuff” in the daily problem sets.


This strategy allows for a wonderfully cross-pollinating atmosphere to arise in whole-class discussion segments. Since everyone has received what they individually needed during the “doing” segments, they are now free to be more open and receptive to what others experienced and discovered while they were lost in their own autonomous worldview.

They can also pay attention to what the instructors really want everybody in the room to experience.

What I value about this strategy is how it turns whole-class discussion segments into resonant, experiential learning sessions for all participants — whatever their starting-point orientation.

Over and over, speed demons are exposed to — and required to notice — the kinds of majestic mathematical discoveries that are possible when you relinquish your foundational belief that only faster can be better.

And katamari experience that “slow and steady” is not an inferior way to approach mathematics but rather, a powerful orientation and set of talents that can reveal mathematical depth and structure that are hidden to the naked speed demon eye.

It also strikes me that much of this is completely at odds with the narrow-minded, and often obtuse insistence in Complex Instruction that everybody always stay together on everything, working on the “same problem” at the “same time.”

I find this obtuse because I have seen how the richness of our human experience comes from coming together and bringing our whole selves to our collaboration — not by holding ourselves back and playing small to avoid making anybody else feel less empowered.

This is what I am trying to get my learners to understand about the value of collaborating with others. Speed demons are rewarded by the mountains they climb and the spectacular landscapes this allows them to experience. Katamari are rewarded by the dazzling richness and microscopic hidden structures they discover. When we bring these experiences together and allow ourselves to share our most powerful insights, that is when we discover the full spectrum of what it means to be mathematical and to be human.